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ELEC 401 – Microwave Electronics
ELEC 401
MICROWAVE ELECTRONICS
Lecture 6
Instructor: M. İrşadi Aksun
Acknowledgements:
1. Materials, illustrations, and some examples have been taken from the
following book:
Modern Microwave Circuits, by N. Kinayman and M. I. Aksun
Artech House , Boston, 2005
M. I. Aksun
Koç University
1/20
ELEC 401 – Microwave Electronics
Outline
 Chapter 1: Motivation & Introduction
 Chapter 2: Review of EM Wave Theory
 Chapter 3: Plane Electromagnetic Waves
 Chapter 4: Transmission Lines (TL)
 Chapter 5: Microwave Network Characterization
 Chapter 6: Smith Chart & Impedance Matching
 Chapter 7: Passive Microwave Components
M. I. Aksun
Koç University
2/20
ELEC 401 – Microwave Electronics
Transmission Lines - General
In general, circuits and networks are made up of
 some components that perform operations on the signals, and
 some others that only carry signals from one place to another
with minimum distortion on the signals.
These components that are responsible for the
transmission of signals are called Transmission Lines
(TL), and they come in various shapes and forms:
M. I. Aksun
Koç University
3/20
ELEC 401 – Microwave Electronics
Transmission Lines - General
Coaxial line
Two-wire line
Strip line
M. I. Aksun
Koç University
Microstrip line
4/20
ELEC 401 – Microwave Electronics
Transmission Lines – General
 For microwave circuits and antennas, TLs play important role,
as apposed to their low-frequency applications.
 TLs are extremely short for low-frequency circuits in terms
of wavelength, and can be modeled by considering only the
bulk resistance, which is close to zero in most cases.
 For microwave and antenna applications, TLs are longer, and
hence their influence on the signals passing through them needs to
be accounted for.
M. I. Aksun
Koç University
5/20
ELEC 401 – Microwave Electronics
Transmission Lines – General
 For the characterization of transmission lines in high-frequency
applications, a special method that utilizes circuit theory as well as
the propagation aspect of the waves is developed and called the
transmission line theory.
 Although the theory can be developed strictly from the
circuit analysis, the major assumptions of the theory, as well
as its limitations, will be missing unless its derivation via
Maxwell’s equations is introduced.
Therefore, a general field-analysis approach is first provided
starting from Maxwell’s equations, and then the transmission line
theory will be developed.
M. I. Aksun
Koç University
6/20
ELEC 401 – Microwave Electronics
Field Analysis of General Cylindrical Waveguides
General Cylindrical Waveguides (WG)
 guide signals in a preferred direction, called the
longitudinal direction (z-direction);
 confine the fields in transverse direction (transverse to
the direction of propagation, the x-y plane).
Assumptions on the structure:
 it is assumed to be uniform in the
direction of propagation; and
y
 it has an arbitrary cross-section in the
transverse direction.
x
z
M. I. Aksun
Koç University
7/20
ELEC 401 – Microwave Electronics
Field Analysis of General Cylindrical Waveguides
WGs vs. TLs:
 TLs are made up of two
conducting materials, while WGs
may consists of only one conductor
or many conductors;
 TLs support a special type of waves called
transverse electromagnetic (TEM) waves,
while WGs can support any type of wave
configuration.
M. I. Aksun
Koç University
8/20
ELEC 401 – Microwave Electronics
Field Analysis of General Cylindrical Waveguides
y
x
z
Observations from the geometry:
 The fields are supposed to be guided along the
z-direction inside the waveguide between the two
conductors;
 the waveguide is uniform and infinite in the
z-direction;
 the walls are assumed to be a perfect conductor,
and
 the medium between the walls is filled with a
dielectric material or air.
Goal is
to find governing equations and field expressions, as it may be possible to find
a simpler version of the wave equation for such a specific class of geometries.
M. I. Aksun
Koç University
9/20
ELEC 401 – Microwave Electronics
Field Analysis of General Cylindrical Waveguides
To achieve the goal, some common features of the geometry may
be utilized in Maxwell’s equations or in the field expressions.
For example, because
 the geometry is uniform and infinite in the
longitudinal direction (z-direction), and
 it is used for the propagation of waves in the zdirection,
the electric and magnetic fields should be in the
following form:
y
x
z
Ex, y, z   Ex, y e jk z z
Hx, y, z   Hx, y e  jk z z
M. I. Aksun
Koç University
10/20
ELEC 401 – Microwave Electronics
Field Analysis of General Cylindrical Waveguides
 jk z
Why Ex, y, z   Ex, y e z ; Hx, y, z   Hx, y e jk z z ?
Because
y
x
 the geometry supports propagation in zdirection, variations in z-direction are
predicted as exponentials with an unknown
factor (propagation constant),
 there is no geometrical variation along the
z-direction, the magnitudes of the fields are
expected to be independent of z.
z
M. I. Aksun
Koç University
11/20
ELEC 401 – Microwave Electronics
Field Analysis of General Cylindrical Waveguides
With this form of the fields, vector quantities in Maxwell’s curl equations
can be written as follows:




  xˆ  yˆ
 zˆ  t  zˆ
x
y
z
z
E  xˆE x  yˆ E y  zˆE z  Et  zˆE z
H  xˆH x  yˆ H y  zˆH z  Ht  zˆH z
Substituting these into Maxwell’s equations results in

zˆ  Et   j Ht
z

t  zˆH z  zˆ  Ht  j Et
z
t  zˆE z 
y
x
z
M. I. Aksun
Koç University
t  Et   jH z zˆ
transverse
longitudinal
t  Ht  jEz zˆ
12/20
ELEC 401 – Microwave Electronics
Field Analysis of General Cylindrical Waveguides
Then, following the manipulations shown below:
Et 
1 


ˆ
ˆ


z
H

z

H
t
z
t
j 
z

t  zˆE z 

zˆ  Et   j Ht
z



 1
zˆ  t  zˆH z  zˆ  Ht 
  j Ht
z
z

 j
making use of zˆ  t  zˆ   t
zˆ  zˆ  Ht   Ht
 2 z 2  k z2
Ht 
1 

ˆ

H

j



z
E
t
z
t
z

k 2  k z2  z
Et 
1 

ˆ

E

j



z
H
t z
t
z
k 2  k z2  z

We obtain
M. I. Aksun
Koç University
t  zˆE z 
13/20
ELEC 401 – Microwave Electronics
Field Analysis of General Cylindrical Waveguides
Let us review what we have achieved:
 Once the longitudinal components of the fields, (Ez ,Hz), are obtained,
the rest of the field components can be calculated from
Et 
1 

ˆ

E

j



z
H
t z
t
z
k 2  k z2  z

Ht 
1 

ˆ

H

j



z
E
t
z
t
z

k 2  k z2  z
Observations: Since Maxwell’s equations are linear in linear media, it is
observed that the fields can be split into three independent modes:
1.Transverse Electric (TE) modes: Ez  0, H z  0
2.Transverse Magnetic (TM) modes: Ez  0, H z  0
3.Transverse ElectroMagnetic (TEM) mode: Ez  0, H z  0
Any field configuration in a waveguide can be written in terms of TE, TM,
and TEM mode contributions, and they are calculated independently.
M. I. Aksun
Koç University
14/20
ELEC 401 – Microwave Electronics
Field Analysis of General Cylindrical Waveguides
TE Waves - Ez  0, H z  0
Et 
1 

ˆ

E

j



z
H
t z
t
z
k 2  k z2  z

Et 
1
2
k  k z2
Ht 
 j t  zˆH z 
1 

ˆ

H

j



z
E
t
z
t
z
k 2  k z2  z

t  Et   jH z zˆ


t  t  zˆH z  k 2  k z2 zˆH z  0
    A    A  2 A
 t  t  zˆH z    t2 zˆH z  k 2  k z2  zˆH z  0


0
Governing equation for TE waves
t2 H z  k 2  k z2 H z  0



k t2
M. I. Aksun
Koç University
15/20
ELEC 401 – Microwave Electronics
Field Analysis of General Cylindrical Waveguides
TM Waves - Ez  0, H z  0
Et 
1 

ˆ

E

j



z
H
t z
t
z
k 2  k z2  z

Ht 
1
2
k  k z2
Ht 
 j t  zˆEz 
1 

ˆ

H

j



z
E
t
z
t
z
k 2  k z2  z

t  Ht  jEz zˆ


t  t  zˆE z  k 2  k z2 zˆE z  0
    A    A  2 A
0
Governing equation for TM waves
M. I. Aksun
Koç University


t t  zˆE z   t2 zˆE z  k 2  k z2 zˆE z  0




t2 E z  k 2  k z2 E z  0



kt2
16/20
ELEC 401 – Microwave Electronics
Field Analysis of General Cylindrical Waveguides
TEM Waves - Ez  0, H z  0
Et 
1 

ˆ

E

j



z
H
t z
t
z
k 2  k z2  z

Et  0, Ht  0
Not if k z  k   
Ht 
is there a trivial solution only?
t  Et  0
Et  t x, y  e  jkz
  Et  0
t2 x, y   0
B.C. on PEC nˆ  Et  0    constant
M. I. Aksun
Koç University
1 

ˆ

H

j



z
E
t
z
t
z
k 2  k z2  z

Governing equation
for TEM waves
17/20
ELEC 401 – Microwave Electronics
Field Analysis of General Cylindrical Waveguides
TEM Waves - Ez  0, H z  0
t2 x, y   0
k z  k   
B.C. on PEC
nˆ  Et  0    constant
Et  t x, y  e  jkz

t  zˆE z  zˆ  Et   j Ht
z

t  zˆH z  zˆ  Ht  j Et
z
transverse

zˆ  Et   j H t
z

zˆ  H t  j Et
z
the coupling of the
electric and magnetic
fields is established
M. I. Aksun
Koç University
18/20
ELEC 401 – Microwave Electronics
Field Analysis of General Cylindrical Waveguides
Salient features of TEM Waves:
 fields of TEM waves are like static fields in the transverse domain,
evidenced by Laplace’s equation in the transverse domain;
t2 x, y   0
there exists a unique scalar potential, in cases of more than one uniform
conductor in the longitudinal direction;
B.C. on PEC
nˆ  Et  0    constant
 the scalar potential is equivalent to voltage in electrostatic fields;
 x, y   V x, y 
M. I. Aksun
Koç University
19/20
ELEC 401 – Microwave Electronics
Field Analysis of General Cylindrical Waveguides
Salient features of TEM Waves (Cont’d):
 although there seems to be no coupling between the electric and magnetic
fields in TEM waves, which is not possible for any wave representation, the
dynamic nature of the fields is taken into consideration by the propagation of
the fields in the z-direction;

zˆ  Et   j H t
z

zˆ  H t  j Et
z
TEM waves don’t suffer from DISPERSION
Phase   t  k z z

dz
 0    kz
0
t
dt
dz 

1
 vp 



dt k z  

M. I. Aksun
Koç University
Independent of frequency
20/20
ELEC 401 – Microwave Electronics
Dispersion
Dispersion causes the shape of a wave pulse to change as it travels
Acoustics and Vibration Animations
Daniel A. Russell, Ph.D., Physics Department, Kettering University
M. I. Aksun
Koç University
21/20
ELEC 401 – Microwave Electronics
Dispersion
A wave packet built up from a sum of 100
cosine functions with different
frequencies. In a non-dispersive medium
all of the different frequency components
travel at the same speed so the wave
function doesn't change at all as it travels.
When the medium is dispersive, the wave
changes shape. For this example, the
dispersion is such that lower frequencies
travel faster than higher frequencies. As a
result, the wave packet spreads out with
the longer wavelengths moving faster and
the shorter wavelengths lagging behind.
Acoustics and Vibration Animations
Daniel A. Russell, Ph.D., Physics Department, Kettering University
M. I. Aksun
Koç University
22/20
ELEC 401 – Microwave Electronics
Transmission Line Analysis via Circuit Theory
Can we model TEM supporting TLs per unit
length by a circuit of lumped components?
y
x
z
i( z, t)
v( z, t )
R z
i( z  Δz, t)
L z
G z
C z
v( z  Δz,t)
z
M. I. Aksun
Koç University
23/20
ELEC 401 – Microwave Electronics
Transmission Line Analysis via Circuit Theory
i( z, t)
v( z, t )
R z
i( z  Δz, t)
L z
G z
C z
v( z  Δz,t)
z
To devise a circuit model for a transmission line, we need to understand the
physical mechanisms involved. Here are the mechanisms and their circuit
models:
 if the conductor of TL has finite conductivity, then there is heat loss, equivalent of a series resistor;
 current flowing on both conductors in opposite directions induces magnetic flux density between
the conductors, which can be modeled as an inductance along the line;
 two conductors with finite surface areas and finite separation distance can naturally be modeled as a
capacitor between the conductors; and finally,
 finite conductivity of the dielectric medium between the two conductors (i.e., the loss) would cause
the current to leak through the media, and modeled as a conductance between the conductors.
M. I. Aksun
Koç University
24/20
ELEC 401 – Microwave Electronics
Transmission Line Analysis via Circuit Theory
What is the governing equations
for the voltage and current on the
transmission line based on the
circuit model of the line?
 Apply Kirchoffs’ voltage and
current laws for the circuit:
i( z, t)
R z
i( z  Δz, t)
L z
G z
v( z, t )
v( z  Δz,t)
z
i z, t 
 Rz i z, t 
t
v z  z, t 
KCL : i z  z, t   i z, t   Cz
 Gz v z  z, t 
t
KVL : v z  z, t   v z, t   Lz
C z
vt , z   ReV z e j t 
i t, z   ReI z e j t 
KVL : V z  z   V z   Lz j I z   Rz I z 
KCL : I z  z   I z   Cz j V z  z   Gz V z  z 
M. I. Aksun
Koç University
25/20
ELEC 401 – Microwave Electronics
Transmission Line Analysis via Circuit Theory
i( z, t)
Rearranging the last two
equations and taking the limit of
result in the following expressions:
R z
i( z  Δz, t)
L z
G z
v( z, t )
C z
v( z  Δz,t)
z
dV z 
  Z I z  Governing
dz
Equations
I z  z   I z 
dI z 


dI
z

 C j V z   G V z 
 Y V z 
z
dz
z  0
dz
where Z and Y are the series impedance and parallel admittance per unit length on the
circuit equivalent of the transmission line, respectively:
V z  z   V z 
dV z 

  L j I z   R I z 
z
dz
z  0
Z  R  j L
M. I. Aksun
Koç University
Y  G  j C
26/20
ELEC 401 – Microwave Electronics
Transmission Line Analysis
i( z, t)
R z
i( z  Δz, t)
L z
I1  I  l 
I z 
I 2  I 0
+
+
V z 
V2  V 0



z l
 , Zc
z 
+
v( z, t )
y
x
G z
C z
v( z  Δz,t)
z
z
dV z 
  Z I z 
dz
dI z 
 Y V z 
dz
d 2V z 
2


V z   0
2
dz
d 2 I z  2
 γ I z   0
2
dz
V1  V  l 
V z   V  e  z  V  e z
I z   I  e  z  I  e z
If R=G=0 (lossless line)
where
  ZY 
   j
M. I. Aksun
Koç University
R  j L  G  j C 
  ZY    2 LC
 j LC  j
27/20
ELEC 401 – Microwave Electronics
Transmission Line Analysis
I1  I  l 
+
V1  V  l 
I z 
+
V z 


z l
 , Zc
dV z 
  Z I z 
dz
I 2  I 0
V
V

V2  V 0
M. I. Aksun
Koç University
I z   I  e  z  I  e z

z 
Define  L 
dV z 
  V  e  z   V  e z   ZI z 
dz
I z  
Define Zc  Z / Y
as the characteristic
impedance of the line
V z   V  e  z  V  e z
+

Z
  z
V e


Z
V  e z
V
V
  ZY
1
1
  z
I z  
V e

V e z
Z /Y
Z /Y
I z  
1   z 1   z
V e
 V e
Zc
Zc
28/20
ELEC 401 – Microwave Electronics
Transmission Line Analysis
Let us consider a special case of lossless line (R=G=0):   j LC  j
V z   V  e  j z  V e j z
I z  
1   j z 1  j z
V e
 V e
Zc
Zc
If we introduce a load ZL
V2
V 0
V  V 
V  V 
1 L
ZL  


 Zc
 Zc
I2
I 0
1 L
V  V 
V  V 

V z   V e
  j z
M. I. Aksun
Koç University
Zc
  Le
j z


L 
V   j z
I z  
e
  Le j z
Zc
Z L  Zc
Z L  Zc

29/20
ELEC 401 – Microwave Electronics
Transmission Line Analysis
Lossless line (Cont’d):
V
V
If we introduce a load ZL
V2
V 0
V  V 
V  V 
1 L
ZL  


 Zc
 Zc
I2
I 0
1 L
V  V 
V  V 

V z   V e
  j z
M. I. Aksun
Koç University
Zc
  Le
j z


L 
V   j z
I z  
e
  Le j z
Zc
Z L  Zc
Z L  Zc

30/20